Introduction

In this lesson, you will learn about exponents and how to read and understand them.

Scripture Connection

Alma 37:6
“Now ye may suppose that this is foolishness in me; but behold I say unto you, that by small and simple things are great things brought to pass; and small means in many instances doth confound the wise.”

Like the small and simple things in this scripture, exponents also bring about great things. They are tiny numbers that make a big difference on the outcome of the answer.

These videos illustrate the lesson material below. Watching the videos is optional,

• Introduction to Exponents (02:36 mins) | Transcript
• Evaluating Numbers with Exponents (02:29 mins) | Transcript

Introduction to Exponents

Remember, multiplication is the shortcut for doing repeated addition:
\begin{align*} 6 + 6 + 6 + 6 + 6& + 6 + 6 + 6 + 6 = 54 \end{align*}

\begin{align*}6 \times 9 & = 54 \end{align*}

Similarly, there is a shortcut for writing multiplication if you do the same thing over and over again:
\begin{align*} 2 \times 2 \times 2 \times 2 \times 2& \times 2 \times 2 = 128 \end{align*}

\begin{align*}2^7 &= 128 \end{align*}

Here 2 is multiplied together 7 times. For the shorthand, write \(2^7 = 128\). That little 7 means the number of times you multiply 2 by itself. It is called an exponent or a power.

During the multiplication lesson, you learned that multiplication is a shortcut for addition. 2 added together 4 times is the same thing as \(2\times4\). Similarly, if you multiply the same number together repeatedly, you can use an exponent. Exponents are little, tiny numbers written near the top right of the number. They may be written smaller, but they have a big impact on the overall answer.

Example 1
\begin{align*}2\times 2\times 2\times 2 = 2^{4} \end{align*}

To calculate this, you can type out the equation as listed above, or you can write it as \(2^{4}\), which is read as “two to the fourth power.” Since the two is raised to the fourth power, you know that you need to multiply two together four times. When you complete this equation, you find that \(2^{4}=16\). Note that this is very different from \(2\times4\), which equals 8. The little number in the exponent position makes a big difference in the answer!

Example 2

\(5^{3}\) tells you to multiply five together three times.
\begin{align*} 5^3 = 5 \times 5 \times 5 = 125 \end{align*}

This equals to 125. Again, this is significantly different from \(5\times3\), which equals 15.

Powers of Ten

Some of the easiest to calculate are the powers of 10. Consider these examples:

• \(10^2 = 100\)

• \(10^4 = 10,000\)

• \(10^8 = 100,000,000\)

Notice the pattern of the total amount of zeros included: it is the same as the exponent.

Evaluating Numbers with Exponents

Anything with an exponent is read as the base number to the power of the number in the exponent position. Using the example from the last section, \(2^{4}\) is read as “two to the fourth power.”

There are two powers that have their own name. Anything raised to the second power is squared, while anything raised to the third power is cubed. In this section, the symbol (\(\cdot \)) indicates multiplication.

Example 3

\(3^{3}\), or three cubed, can also be written as \(3\cdot 3\cdot3\). Three multiplied by three equals nine, and nine multiplied by three equals 27, so \(3^{3}=27\).

Example 4

\(2^{3}\), or two cubed, can be written as \(2\cdot 2\cdot 2\). Two times two is four, and four multiplied by two is eight, so \(2^{3}=8\).

Example 5

\(7^{2}\), or seven squared, is the same as \(7\times7\), so \(7^{2}=49\).

Example 6

\(1^{14}\) uses 1 as the base number. One to the fourteenth power is one multiplied by one repeated 14 times, which equals one.

\begin{align*}1^{14} &= 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1 = 1 \\\end{align*}

One to any power is equal to one.

Things to Remember

• Just as multiplication is a shortcut for addition, exponents are a shortcut for multiplication.
• Anything to the second power is squared, while anything to the third power is cubed.
• One raised to any power will always equal one.

Practice Problems

1. \(1^2\, = \,?\) (
   Solution

   x

   Solution: 1
   Details: One raised to any power is equal to one. In this example \(1^2 = 1 \times 1 = 1\)
   )
2. \(8^2\, = \,?\) (
   Video Solution

   x

   Solution: 64
   Details:
   

   
   (Introduction to Exponents #2 (00:27 mins) | Transcript)
   | Transcript)
3. \(0^3\, = \,?\) (
   Solution

   x

   Solution: 0
   Details:
   As you've seen in multiplication, any number multiplied by zero is 0:
   \(0^3 = 0 × 0 × 0 = 0\)
   )
4. \(5^3\, = \,?\) (
   Solution

   x

   Solution: 125
   Details:
   \(5^{3}\) means 5 is being multiplied by itself 3 times.
   
   \(5 \times 5 \times 5\)
   
   Since everything is being multiplied together, you can start on the left and move right, doing one operation at a time.
   
   
   \((5 \times 5) \times 5 =\)
   
   \(25 \times 5 =\)
   
   The final answer is: 125.
   )
5. \(4^3\, = \,?\) (
   Video Solution

   x

   Solution: 64
   Details:
   

   
   (Introduction to Exponents #5 (00:36 mins) | Transcript)
   | Transcript)
6. \(3^4\, = \,?\) (
   Solution

   x

   Solution:
   81
   )

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